In my previous post I was discussing the Conway polynomial. In the early
1980s, Vaughan Jones came up with a shockingly similar knot invariant.
Interestingly, he arrived at it via von Neumann algebras, a kind of
operator algebra invented by von Neumann to study the foundations of
quantum physics (among other things). With the Jones
polynomial and Conway polynomial around people immediately started
searching for a generalization that would encompass both and soon a whole
bunch of people found one. It's often called the HOMFLY polynomial these
days after the initials of some (but not all) of its discoverers. This
``polynomial'' is really a Laurent polynomial in two variables, namely a
polynomial in , , and . It may be calculated by the
following rules:

Rule 1: If is the unknot (an unknotted circle), . This is sort of a normalization rule.

Rule 2: Suppose , , and are 3 knots (or links)
differing at just one crossing (we're supposing them to be drawn as
pictures in 2 dimensions).

At this crossing they look as follows:

looks like:

looks like:

looks like:

(Again, all of them should have
arrows pointing down, and any rotated version of this picture is fine too.)

Then we have the ``skein relation'' (here's where it's different
from the Conway polynomial):

Nice and simple! (Though it's not at all trivial to prove its existence
and uniqueness!) Note that the variable keeps track of handedness or
what a physicist might call ``parity'' -- it's an easy exercise to show
that if the knot is the mirror image of then

It's a pleasant exercise to use rules 1) and 2) to calculate this
polynomial for the left-handed and right-handed trefoil knot and see
that the answers are different.

Now the question becomes, what is this polynomial trying to tell us?
Here I need a digression into the theory of braids, and, to keep the
physicists interested, I'll begin with anyons.

We all know that particles come in two fundamentally different flavors:
bosons and fermions. The argument for this is simple, well-known, and a
bit misleading. It goes like this. Say we have a bunch of identical
particles. Their state is described by a wave vector (a vector in a
complex Hilbert space). We may permute the particles without really
doing anything since they are identical, you can think of it as just
permuting their ``labels''. Now in quantum mechanics two wave vectors
which differ only by a phase (a scalar factor of unit modulus) describe
the same physics. Thus we must have a representation of the symmetric group
on the Hilbert space, and it must map any permutation to a scalar
multiple of the identity. There are only 2 such representations (another
charming exercise): the trivial representation and the one mapping each
permutation to its sign. In the former case we say the particles are
bosons, and in the latter, fermions.

This seems to be the case in reality. Interestingly, all the
fundamental particles one might call ``matter'' are fermions (quarks and
leptons), while all the particles one might call ``force fields'' are
bosons (the photon, W, Z, and gluons). Here of course I am skirting the
issue of the Higgs particle, that curious fudge factor. If the Japanese
decide to pay for the superconducting supercollider we will see if the
Higgs exists.

If one pays close attention to the argument, however, it's full of holes.
First of all, why do we really need a representation of -- in
quantum mechanics a projective representation is good enough! Secondly, if
one considers representations where does not act as scalars but as an
``internal symmetry group'' one gets even more possibilities. These were
investigated under the name of parastatistics. Anyway, one can come up
with a better argument, the spin-statistics theorem, in relativistic
quantum field theory, and that, together with the fact that parastatistics
can be redescribed as fermions and bosons in disguise, seems to give a
solid explanation for why all we see is bosons and fermions. (Though I
couldn't say I'm very familiar myself with the whole story.)

Now for the catch: the spin-statistics theorem only holds for spacetimes
of dimension 4 and up. You could just say ``thank heavens! that just
happens to apply to our universe!'' and leave it at that, or you could
note that it's occaisionally possible to simulate universes of lower
dimension. Take, for example, a thin 2-dimensional layer of stuff: this
can act like a little 3-dimensional spacetime. Similarly, filaments can
act like 2-dimensional universes. These days condensed matter theorists
delight in the odd processes that occur in these contexts, and it was
only a matter of time before someone noted that one can, at least in
principle, arrange to get particles that are neither bosons or fermions.
Wilczek is generally credited with taking the idea of these ``anyons''
seriously, though it had occured to others earlier.

Here's how it goes in its most primitive form. Say we have some tubes of
magnetic flux moving around. (One can play with these flux tubes using
superconductors, for example). As long as these tubes stay parallel the
problem is essentially a 2-dimensional one: pick a plane perpendicular to
the flux tubes and just pay attention to their intersection with that
plane. Each tube intersects the plane in a spot which we will regard as a
``particle''. In each spot there is a B-field perpendicular to the
plane, and going around each spot is an A-field whose curl is the
B-field. If you like you can think of each spot as a ``vortex'' of
the A-field. Now suppose -- and here I don't know if this has ever
been experimentally achieved -- that each tube is electrically charged. In
our planar picture then, we've got these ``particles'' which are charged,
each also being a vortex of the A-field. Let's assume that all these
particles are identical. Now let's see what happens when we interchange
two of them. Recall that when you move a particle with charge through
an A-field, its phase changes by

where is the line integral of the A-field along the path
traversed by the particle. Thus when we interchange two of our particles
-- and here I mean you physically ``grab'' them and move them around each
other so that they trade places! -- the wave vector of the system is
changed by a phase. I'll let you calculate it. The point is, depending on
the charge and the strength of the magnetic flux tube, one can arrange for
this phase to be whatever one likes -- any complex number of magnitude
one! If this number equals one has bosons, if it's one has
fermions, but otherwise one has anyons.

One can have fun playing with this idea. Many people have. Wilczek is
a big proponent of an anyonic theory of high-temperature
superconductivity, although recent experiments, demonstrating an
apparent absense of spontaneous parity violation that one would expect
in this theory, seem to rule it out. It's not 100% dead, though, and in
my opinion it's so beautiful that someone should try to make a
superconductor based on this principle just for the glory of it.
Something that anyone who has followed me so far can have fun doing, is
to see what sort of particle a bound state of anyons acts like. It's
well-known that two fermions together act like a boson, two bosons act
like a boson, and a boson and a fermion act like a fermion ...extend
this to anyons.

A question for the real physicists out there: has anybody ever really
made anyons and played with them yet?

Now anyons have a lot to do with braids because, as you may have
noticed, I have covertly stopped thinking of the the operation of
interchanging identical particles as an abstract ``switching'' -- modelled
by the symmetric group -- and started thinking of it as moving one
particle around another. If one draws the worldlines of some anyons
as one moves them around each other this way, one has -- a braid! I.e.,
out with the symmetric group, in with the braid group!

The high-handed manner in which I've thrown out the symmetric group and
started working with braid group statistics should disturb you, but
again I can cite fancy mathematical physics papers which should allay
your fears. A very nice one is ``Local Quantum Theory and Braid Group
Statistics'' by Froehlich and Gabbiani, which gives a proof of the
generalized spin-statistics theorem that holds in 2 and 3 dimensions.

So now we see a close relation between quantum theory -- to be precise,
``statistics'' in quantum theory -- and the braid group. Looking back at
Kauffman's original insight into the relation of knots and quantum
mechanics, it's not blindingly obvious what that has to do with
this! Nonetheless it's all part of one story. (A story which I
strongly feel is far from over.) More later.